shortest paths - university of washingtonshortest paths 3 recall path cost ,path length t c•phsota...
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Shortest Paths
CSE 373Data Structures
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Readings
• Reading Chapter 13› Sections 13.5 to 13.7
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Recall Path cost ,Path length• Path cost: the sum of the costs of each edge• Path length: the number of edges in the path
› Path length is the unweighted path cost
Seattle
San FranciscoDallas
Chicago
Salt Lake City
4
2 2
23
2 23
length(p) = 5cost(p) = 11
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Shortest Path Problems• Given a graph G = (V, E) and a “source” vertex s
in V, find the minimum cost paths from s to every vertex in V
• Many variations:› unweighted vs. weighted› cyclic vs. acyclic› pos. weights only vs. pos. and neg. weights › etc
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Why study shortest path problems?
• Traveling on a budget: What is the cheapest airline schedule from Seattle to city X?
• Optimizing routing of packets on the internet:› Vertices are routers and edges are network links with
different delays. What is the routing path with smallest total delay?
• Shipping: Find which highways and roads to take to minimize total delay due to traffic
• etc.
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Unweighted Shortest PathProblem: Given a “source” vertex s in an unweighted
directed graph G = (V,E), find the shortest path from s to all vertices in
G
A
C
B
D
F H
G
E
Source
Only interested
in path lengths
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Breadth-First Search Solution
• Basic Idea: Starting at node s, find vertices that can be reached using 0, 1, 2, 3, …, N-1 edges (works even for cyclic graphs!)
A
C
B
D
F H
G
E
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Breadth-First Search Alg.• Uses a queue to track vertices that are “nearby”• source vertex is s
Distance[s] := 0Enqueue(Q,s); Mark(s)//After a vertex is marked once
// it won’t be enqueued againwhile queue is not empty do
X := Dequeue(Q);for each vertex Y adjacent to X do
if Y is unmarked thenDistance[Y] := Distance[X] + 1;Previous[Y] := X;//if we want to record pathsEnqueue(Q,Y); Mark(Y);
• Running time = O(|V| + |E|)
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Example: Shortest Path length
A
C
B
D
F H
G
E
0
Queue Q = C
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Example (ct’d)
A
C
B
D
F H
G
E
0
Queue Q = A D E
1
1
1
Previouspointer
Indicates the vertex is marked
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Example (ct’d)
A
C
B
D
F H
G
E
0
Q = D E B
1
1
1
2
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Example (ct’d)
A
C
B
D
F H
G
E
0
Q = B G
1
1
1
2
2
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Example (ct’d)
A
C
B
D
F H
G
E
0
Q = F
1
1
1
2
2
3 4
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Example (ct’d)
A
C
B
D
F H
G
E
0
Q = H
1
1
1
2
2
3
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What if edges have weights?
• Breadth First Search does not work anymore › minimum cost path may have more edges than
minimum length path
A
C
B
D
F H
G
E
2 3
2 1
1
42
11
93
8
3
Shortest path (length)from C to A:C A (cost = 9)
Minimum Cost Path = C E D A(cost = 8)
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Dijkstra’s Algorithm for Weighted Shortest Path
• Classic algorithm for solving shortest path in weighted graphs (without negative weights)
• A greedy algorithm (irrevocably makes decisions without considering future consequences)
• Each vertex has a cost for path from initial vertex
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Dijkstra’s Algorithm
• Edsger Dijkstra(1930-2002)
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Basic Idea of Dijkstra’sAlgorithm (1959)
• Find the vertex with smallest cost that has not been “marked” yet.
• Mark it and compute the cost of its neighbors.• Do this until all vertices are marked.• Note that each step of the algorithm we are
marking one vertex and we won’t change our decision: hence the term “greedy” algorithm
• Works for directed and undirected graphs
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Dijkstra’s Shortest Path Algorithm
• Initialize the cost of s to 0, and all the rest of the nodes to ∞
• Initialize set S to be ∅› S is the set of nodes to which we have a shortest path
• While S is not all vertices› Select the node A with the lowest cost that is not in S
and identify the node as now being in S› for each node B adjacent to A
• if cost(A)+cost(A,B) < B’s currently known cost– set cost(B) = cost(A)+cost(A,B)– set previous(B) = A so that we can remember the path
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Example: Initialization
v1
v7v6
v2
v5v3 v4
4 1
2
103
64
22
85
1
0 ∞
∞ ∞
∞
Pick vertex not in S with lowest cost.
∞ ∞
Cost(source) = 0 Cost(all vertices but source) = ∞
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Example: Update Cost neighbors
v1
v7v6
v2
v5v3 v4
4 1
2
103
64
22
85
1
0 2
∞ ∞
1
∞ ∞
Cost(v2) = 2Cost(v4) = 1
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Example: pick vertex with lowest cost and add it to S
Pick vertex not in S with lowest cost, i.e., v4
v1
v7v6
v2
v5v3 v4
4 1
2
103
64
22
85
1
0 2
∞ ∞
1
∞ ∞
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Example: update neighbors
v1
v7v6
v2
v5v3 v4
4 1
2
103
64
22
85
1
0 2
3 3
1
9 5
Cost(v3) = 1 + 2 = 3 Cost(v5) = 1 + 2 = 3 Cost(v6) = 1 + 8 = 9 Cost(v7) = 1 + 4 = 5
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Example (Ct’d)
v1
v7v6
v2
v5v3 v4
4 1
2
103
64
22
85
1
0 2
3 3
1
Pick vertex not in S with lowest cost (v2) and update neighbors
9 5
Note : cost(v4) not updated since already in S and cost(v5) not updated since it is larger than previously computed
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Example: (ct’d)
v1
v7v6
v2
v5v3 v4
4 1
2
103
64
22
85
1
0 2
3 3
1
Pick vertex not in S (v5) with lowest cost and update neighbors
9 5No updating
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Example: (ct’d)
v1
v7v6
v2
v5v3 v4
4 1
2
103
64
22
85
1
0 2
3 3
1
Pick vertex not in S with lowest cost (v7) and update neighbors
8 5
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Example: (ct’d)
v1
v7v6
v2
v5v3 v4
4 1
2
103
64
22
85
1
0 2
3 3
1
6 5
Pick vertex not in S with lowest cost (v7) and update neighbors
Cost(v6) = min (8, 5+1) = 6
Previous cost
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Example (end)
v1
v7v6
v2
v5v3 v4
4 1
2
103
64
22
85
1
0 2
3 3
1
Pick vertex not in S with lowest cost (v6) and update neighbors
6 5
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Data Structures• Adjacency Lists
1234567
2 2G
0∞∞∞∞∞∞
C4 1
v1
v7v6
v2
v5v3 v4
4 1
2
103
64
22
85
1
4 3 5 101 4 6 53 2 5 27 6
6 1
6 8
7 4
nextcost
adj
Priority queue for finding and deleting lowest cost vertexand for decreasing costs (Binary Heap works)
P Q
previous cost priority queue pointers
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Priority Queue
v1
v7v6
v2
v5v3 v4
4 1
2
103
64
22
85
1
0 2
∞ ∞1
∞ ∞
2
5 7
3 6
02∞1∞∞∞
C
14
253
Q1234567
node number1
2 3
54
index in heap
1
1
Before the update, butafter find min.,i.e., v1 and v4have been “deletemin”
This is somewhat arbitrary and depends when the heap was first built
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Priority Queue
v1
v7v6
v2
v5v3 v4
4 1
2
103
64
22
85
1
0 2
3 ∞1
∞ ∞
2
5 7
3 6
0231∞∞∞
C
14
253
Q1234567
node number1
2 3
54
index in heap
141
update node 3decreasecost
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Priority Queue
v1
v7v6
v2
v5v3 v4
4 1
2
103
64
22
85
1
0 2
3 ∞1
∞ ∞
2
3 7
5 6
0231∞∞∞
C
12
453
Q1234567
node number1
2 3
54
index in heap
141
percolate up
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Time Complexity
• n vertices and m edges• Initialize data structures O(n+m)• Find min cost vertices O(n log n)
› n delete mins• Update costs O(m log n)
› Potentially m updates• Update previous pointers O(m)
› Potentially m updates• Total time O((n + m) log n) - very fast.
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Correctness
• Dijkstra’s algorithm is an example of a greedy algorithm
• Greedy algorithms always make choices that currently seem the best› Short-sighted – no consideration of long-term or global
issues› Locally optimal does not always mean globally optimal
• In Dijkstra’s case – choose the least cost node, but what if there is another path through other vertices that is cheaper?
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THE KNOWNCLOUD
G Next shortest path from inside the known cloud
P
“Cloudy” Proof
• If the path to G is the next shortest path, the path to P must be at least as long. Therefore, any path through P to G cannot be shorter!
Source
Least cost node
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Inside the Cloud (Proof)
• Everything inside the cloud has the correct shortest path
• Proof is by induction on the number of nodes in the cloud:› Base case: Initial cloud is just the source with
shortest path 0› Inductive hypothesis: cloud of k-1 nodes all have
shortest paths› Inductive step: choose the least cost node G
has to be the shortest path to G (previous slide). Add k-th node G to the cloud